Introduction

A standard assumption in cartel formation and collusion sustainability has been that all firms are identical in terms of their costs, even if there was a degree of differentiation amongst their products or in firm’s timing decision. However, the assumption of cost symmetry is unrealistic and very restrictive when one looks to real markets. A primer approach to tackle cost asymmetries was inferred from an early paper by ^{Patinkin (1947}), where a cartel maximises total industry profits and therefore allocates output quotas so that the cartelised industry operates as if it was a multiplant monopolist allocating output between plants. Moreover, costs asymmetries are proved to play also an important role when firms attempt to reach collusion^{1}. As ^{Bain (1948}) points out cost heterogeneity would mean that, in the absence of side payments between firms, such an allocation might not be viable as inefficient firms may obtain lower profits in the cartel than in the non-cooperative equilibrium. The intuition is that firms may find it difficult to agree on a common collusive policy because firms with a lower marginal cost will insist in lower prices than those firms with higher marginal cost would wish to sustain. More generally, the common wisdom is that the diversity of cost structures may rule out any possible agreement in pricing policies and so exacerbate coordination problems. In addition, technical efficiency would require allocating higher production quotas to low-cost firms, but this would clearly be difficult to sustain in the absence of explicit agreements or side transfers. Thus, it seems natural to characterise a class of agreements different to the most collusive outcome (i.e., the monopoly solution), namely imperfect collusion^{2}. Such an agreement allows firms to achieve some degree of collusion and also to sustain that agreement over time.

Admittedly, the possibility of collusive firms operating with different cost functions has also received some attention in the modern Industrial Organisation literature. For instance, ^{Osborne and Pitchik (1983}) in a static non-cooperative model where firms are capacity-constrained allow for side payments and show that the profit per unit of capacity of the small firm is higher than that of the large one. ^{Schmalensee (1987}) in a static game with linear costs in a Cournot setting characterises the set of profit vectors by applying a number of selection criteria such as the Nash bargaining solution. He finds that if a leading firm’s cost advantage is substantial, its potential gains from collusion are relatively small. By their very nature, however, in a static model cartel members do not cheat on a cartel agreement since it is assumed that agreements are sustained through binding contracts. This may, therefore, be viewed as a model of explicit or binding collusion. These papers thus do not impose the incentive compatibility constraints of subgame perfection and the collusive outcome derived in their models may not be self-enforced. Looking at ^{Friedman’s (1971}) supergame- theoretic approach to collusion a few papers have also considered the problem of enforcement of collusive behaviour with asymmetric firms. ^{Rothschild (1999}) shows that the stability of the cartel may depend crucially upon the relative efficiencies of the firms and that joint profit maximisation becomes less likely as cost functions differ across firms. ^{Vasconcelos (2005}) in a quantity setting oligopoly model assumes asymmetry by modelling that firms have different shares of a specific asset and shows that the sustainability of perfect collusion crucially depends on the most inefficient firm in the agreement, which represents the main obstacle to the enforcement of collusion. More recently, ^{Miklós-Thal (2011}) shows that in a Bertrand supergame some collusion is also sustainable under cost asymmetry whenever collusion is sustainable under cost symmetry and ^{Contreras et al. (2008}) have shown that with differentiated products a cartel may also be stable provided that returns to scale are high enough. Summarising, both the literature on static cartel stability and the dynamic models of tacit collusion suggest that collusion is unlikely to be observed in the presence of substantial competitive advantage, and therefore, a prior step before studying collusion sustainability when costs are heterogeneous and firms agree on output quotas is to consider whether collusion is viable. In line with these results, the analysis of the empirical literature also indicates that cost asymmetries hinder collusion (see for instance ^{Levenstein and Suslow, 2006})^{3}.

In this paper, we investigate the extent to which imperfect collusion can be a way to sustain a collusive agreement in the presence of large cost asymmetries when firms produce a homogeneous product. First, we study whether cost heterogeneity is sufficient to make it impossible for firms to collude (and its sustainability over an infinite horizon) when coordination is not necessarily on the allocation that maximises total industry profits. In this sense, in their empirical studies ^{Eckbo (1976}) and ^{Griffin (1989}) provide an interesting motivation with this respect by finding that even though cartels that are made up of similar-sized firms are more able to raise prices, in some cases high-cost members of a cartel may produce at a cost larger than 50% above low-cost members^{4}. Consequently, the question about why and to what extent should cost asymmetries be a re-straint for collusion naturally arises. Secondly, we also study how the degree of collusion can be endogenously determined. Such degree of collusion may be interpreted as part of an explicit cartel agreement (and allocating thus different output quotas depending on the firm’s relative efficiency) or it can be thought of as a degree of coordination when collusion is tacit.

Once the degree of collusion is characterised we investigate the sustainability of imperfect collusive agreements in a multi-period duopoly model. We use subgame perfect Nash equilibria -henceforth, SPNE- as solution concept. It is well known that this repeated game setting exhibits multiple equilibria. To select among those equilibria we adopt the particular criterion of restricting strategies to grim trigger strategies where firms adhere to the collusive agreement until there is a defection, in which case they revert forever to the static Cournot equilibrium. The key feature is the assumption that firms maximise the summation of its own profits plus a proportion of the profits of the other firms. As a consequence, this proportion may be considered as the degree of coordination under an (imperfect) collusive agreement. This approach has received growing attention of scholars (see for instance ^{Symeonidis, 2008} or ^{Matsumura et al., 2013}), it is also closely related to the coefficient of cooperation defined by ^{Cyert and DeGroot (1973}), and captures the relative performance approach that is evolutionary stable (^{Vega-Redondo, 1997}). These objective functions are also in line with the growing and more recent behavioural economics literature, as well as with experimental games that test the extent to which subjects are concerned with reciprocity (see for instance ^{Fehr and Schmidt, 1999} and ^{Charness and Rabin, 2002} respectively).

Our main contribution is to show that in a quantity-setting model some degree of imperfect collusion can always be sustained regardless of the cost heterogeneity. The intuition is that even though cost asymmetry hinders collusion, for each possible level of their discount factor firms can always coordinate on an output level below the competitive one. Hence, one can expect collusion between firms to occur, at least to some extent, also with very asymmetric firms since these firms may still have an incentive to adhere to an imperfectly collusive agreement. Consequently, the antitrust authorities should also be cautious when firms in an industry have significantly different cost functions because firms’ willingness to collude may be still present. It is also obtained that the endogenous degree of collusion to be sustained has an upper bound that can never be overcome. This boundary is determined by the most inefficient firm and depends negatively on the degree of cost asymmetry. In other words, although some degree of coordination (generally) might lead to higher profits overall, at the same time the aforementioned adjustments in the degree of collusion seem to be limited by the willingness of the most inefficient firms. Therefore, in a market with significant production cost asymmetries, a negative relationship between firms’ efficiency and the endogenous choice of the degree of sustainable collusion can be expected^{5}.

The remainder of the paper is organised as follows. In Section 2 the model is presented. In Sections 3 and 4 we analyse imperfect collusion and its sustainability. In Section 5 the degree of collusion is made endogenous. Section 6 presents some extensions of the model showing, for instance, that the effect of cost asymmetries on firms’ collusion incentives is also robust to other ways to parameterise the product-market competition. Section 7 concludes. All proofs are grouped together in the Appendix.

Set Up of The Model

We consider an industry with two asymmetric firms indexed by *i* = 1,2 where each firm simultaneously produces a homogeneous product. Even though quantity competition is assumed, it is well-known from ^{Kreps and Scheinkman (1983}) that in a two-stage oligopoly game where, first, there is simultaneous production, and, second, after production levels are made public, there is price competition, the unique equilibrium outcome is the Cournot outcome. Consequently, the present model could also be interpreted as a market in which firstly capacities are determined and secondly price competition takes place. Regarding production costs, we assume a technology such that firms produce with a quadratic cost function
where *qi* is the output produced by firm *i*. The function ρ*i* (*c*) accounts for the asymmetry between firms, taking values 1 + *c* and 1 - *c*, respectively, with 0 < *c* < 1. Without loss of generality, we assume that firm 1 is the inefficient and firm 2 is the efficient one^{6}. Hence, as *c* approaches the unity firms become more asymmetric. Differences in the slope of marginal costs can be interpreted for instance as resulting from differences in capital stocks^{7}. Some features of the cost functions should be emphasised at the outset. Firstly, following the reasoning provided by ^{Rothschild (1999}) and in order for the collusive outcome not to become trivial, the cost functions are not linear. If on the contrary, firms had different but constant marginal costs, it would clearly be practical for the firm with the lowest costs to produce the entire output. In the present model though, a switch of production exclusively to the most efficient firm would raise industry costs. The second feature is that fixed costs are taken to be zero. This simplifying assumption is common in the literature, partly for simplicity but also because provided that fixed costs are not so high as to force firms out of the market, the relative magnitudes of the payoffs to firms from different actions are unaffected by the omission of a fixed cost.

The industry inverse demand is given by the piecewise linear function
, where *p* is the output price and Q=+
is the industry output. In the absence of coordination, each firm plays a Cournot stage game. The profit function for firm *i* is given by:

We characterise imperfect collusion in the Cournot-stage game considering a particular model where each firm maximises the sum of its own profit and a fraction of the profit of the other firm. Explicitly, each firm i maximises
where
. We assume α to be constant and symmetric in such a way that, regardless of whether a firm is efficient or inefficient their degrees of reciprocal concern with the rival coincide. The parameter α thus can be interpreted as representing the degree of collusion^{8}. Admittedly, although we build a direct link between α and the degree of collusion, the reader might feel more comfortable when such link is made explicit and based on a direct behavioural assumption like the output produced. Arguably, we could also interpret our model as one in which firms’ strategy set is a quantity in the interval between the joint-profit maximising allocation and the asymmetric Cournot equilibrium where α merely parameterises the most collusive output achievable as a result of the efficiency differences between firms.

Definition 1: Collusion is said to be imperfect if . On the contrary, collusion is said to be perfect if α = 1.

An alternative consideration is that since we assume away side-payments, one could think that firms should bargain over possible outputs^{9}. Our approach is somewhat different. We assume that firms coordinate to behave as in a model of symmetric cross-ownership but according to their different efficiency level, even though in the present model firms behave as Cournot competitors when they do not collude. In other words, we consider a profit-sharing rule where firms’ profits are to some extent proportional to capital stocks. ^{Bos and Harrington (2010}) and the references cited therein provide abundant motivation about this rule often referred to as a proportional rule.

Characterisation of an Imperfect Collusive Agreement

Let qic denote the quantity corresponding to the collusive output. Given a degree of asymmetry c, we obtain the following collusive equilibrium quantities and profits for firm i,

where throughout the paper and abusing notation, we assume as exogenously given c. It is straightforward to obtain the Nash-Cournot non-cooperative equilibrium and the associated level of profits for each firm in (2) for α = 0. We denote them by qi*, and ∏i* respectively.

We note that
since if α increases firms’ production tends to the one of a perfectly collusive market where outputs are reduced in order to increase the price. However, qic(α) decreases (increases) with c for firm 1 (firm 2) since collusion requires an efficient reallocation of outputs concentrating production in the more efficient firm. Consequently, when c increases over a certain critical value, an inefficient firm may not be interested in being part of a perfectly collusive agreement because profits attained under such an agreement are lower than those obtained at the Cournot stage game. Intuitively, if collusion is perfect, in order to maximise joint profits the reduction needed in the quantity produced by the inefficient firm compared to the Cournot equilibrium is not compensated by the price increase. This argument also applies if α is large enough but lower than 1. However, if α is low enough imperfect collusion can still allow the inefficient firm to obtain larger profits than without collusion for any level of efficiency differences. On the contrary, it is straightforward to check that
for any
. Since collusion implies that the cartel minimises the total costs of producing a given output level, allocating thus production amongst the firms in such a way that marginal costs are equalised, the efficient firm always benefits from collusion. In fact, this implies that a firm’s share in the output of the cartel, and the profits which it obtains are larger if the firm is relatively more efficient^{10}. The following lemma proves that at least some collusion is always feasible with an upper bound negatively associated with the cost asymmetry^{11}.

Lemma 1: For any
there always exists
such that
if
, where *decreases with c. Conversely,*
*if*
*.*

Sustainability of Imperfect Collusion

We assume in this section that firms compete repeatedly over an infinite horizon with complete information (i.e. both firms observe the whole history of actions) and discount the future according to a common discount factor
At any stage the profit function is given by (1). Time is discrete and dates are denoted by *t =* 1, 2,... In this framework, a pure strategy for firm *i* is an infinite sequence of functions
with
where
is the set of all possible histories of actions (output choices) of each firm up to *t* - 1, with

typical element
, I = 1,2, τ = 1, …, t-1, and *Q* is the set of output choices available to each firm. We follow ^{Friedman (1971}) restricting our attention to the case where each firm is only allowed to follow grim trigger strategies such that firms adhere to a collusive agreement until there is a defection, in which case they revert forever to the static Nash-Cournot equilibrium. Hence,
can be specified as follows. At *t =* 1, _{
Si
}
^{1} = _{
qi
}
^{c}, while at *t =* 2,3,...

Although there is a multiplicity of equilibria since the above strategies sustain different collusive outputs, we focus on an equilibrium that depends on α and hence on the efficiency differences between firms^{12}. Thus, firms producing qic(α) in each period can be sustained as a SPNE of the repeated game with the strategy profile (3) if and only if the following conditions are satisfied

where denotes the profits obtained by firm i in an optimal deviation from the collusive output qic(α). In other words, if δ exceeds a certain critical level the inequalities described in (4) are satisfied. We denote by this critical value of the discount factor where qic(α) is a SPNE of the repeated game if . In order to characterise , we need to define which is obtained by replacing in for where . Therefor, , where

As we prove in the proof of Proposition 1 the condition on δ in (4) is always more easily satisfied for the efficient firm than for the inefficient one (namely,
)^{13}. Therefore, we can define collusion sustainability as follows:

Definition 2: Imperfect collusion is sustainable if .

The following proposition shows that (some) imperfect collusion is always sustainable regardless of the cost asymmetry of firms.

Proposition 1: For any there always exists such that no matter how small δ is, collusion on an output level below the competitive one is sustainable if with . Furthermore, increases with α and c.

In other words, α is an upper bound on the degree of collusion that can be sustained in the infinitely repeated game. The above proposition thus extends Lemma 1 in the sense that collusion sustainability is only possible if firms agree on a lower degree of collusion^{14}. The intuition is fairly simple. From ^{Rothschild (1999}) we know that if in our model α = 1, perfect collusion becomes harder to sustain if firms’ cost asymmetry increases in such a way that if c is large enough, perfect collusion is not sustainable^{15}. However, if we allow firms to sustain less collusion, the agreement may be sustained. Consequently, there is always a small enough degree of collusion that can be sustained despite the difference between firms’ costs. A numerical example may help clarify our result. For instance, perfect collusion with c > 0.202 yields to
. Consequently, the standard joint profit maximisation allocation cannot be sustained. However, when we consider imperfect collusion, α ≤ 0.5 can be sustained if δ ≥ 0.467 (since
). Lemma 1 and Proposition 1 show that if α is small enough, the standard Prisoner’s Dilemma scheme of a quantity-setting collusive market is restored since
where (some) imperfect collusion is always sustainable. This result can also be interpreted in the following way.

One can check that for a given value of δ and when α = 0,

The incentive constraint described in (4) is binding for the inefficient firm:

Hence, there always exists an incentive to assume at least some coordination between firms because, in a Cournot market, collusion profits increase with the degree of collusion more than what deviation profits do. In fact, for a given the left hand side of the inequality , which is obtained from (4), has an inverse U-shape function relationship with respect to α. Figure 1 shows this insight for δ = {0.6, 0.75, 0.9} and c = 0.25 where .In these cases, the maximum level of imperfect collusion that can be sustained as a SPNE of the infinitely repeated game is respectively α = {0.559, 0.684, 0.801}.

Endogenous Imperfect Collusion

In this section, α is made endogenous by adding an initial stage in which firms choose the extent of imperfect collusion. We assume that firms firstly and simultaneously choose a degree of collusion, namely αi, to afterwards continue with the infinitely repeated game described above. We characterise the problem that firms solve in the first stage denoting the profits that firms obtain in the initial stage by for i and j = 1, 2 and i ≠ j where firm i maximises

and where
is analogous to the collusive profit function defined in (2) but for the case of asymmetric α. Firm i chooses αi anticipating that both firms will only maximise their (imperfectly) collusive profits whenever the degree of collusion chosen by firms is sustainable. Otherwise, firms’ profits correspond with the non-cooperative Nash-Cournot equilibrium. In other words, once firms decide about αi and αj, and according to (3), imperfect collusion may be sustained as a SPNE of the repeated game only if
. On the contrary, if
imperfect collusion cannot be sustained. Consequently, the solution to (5) gives rise to two different reaction functions αi (αj) for i and j = 1, 2 and i ≠ j for each possible value of δ. Therefore, the intersection of these reaction functions leads us to a solution with different values for the endogenous degree of collusion for firms. We note that even though firms might deviate from the output agreed, we implicitly assume that firms cannot deviate from the degree of collusion decided in the first stage as long as αi and αj are non-cooperatively decided in the first stage^{16}.

We assume for simplicity though that firms agree on a common value (such that αi=αj=α) that we denote by α* and that we can interpret as an upper bound on the degree of collusion to be sustained^{17}. In order to obtain α*, we firstly analyse how the functions defined in (2) change with α^{18}.

Lemma 2: The function defined in (2) always increases with α in the interval . Conversely, increases with α up to where it reaches a maximum value.

The most efficient firm always prefers more collusion whereas for the inefficient firm, if the rule for output allocation in order to (imperfectly) collude implies that its production is highly reduced and its profits are smaller than when . On the contrary, if inefficient firm’s profits increases with α and, therefore, this firm would rather choose the highest possible (namely, sustainable) value of α over the interval . Consequently, a potential mutual agreement on α to which both firms adhere may be obtained since it follows directly from Lemma 2 that the inefficient firm’s decision on α is binding in order to sustain collusion.

Proposition 2: Let’s consider the function . Then, if the endogenous degree of collusion is such that . On the contrary, for a given , the endogenous degree of collusion α* is the α that solves the equation .

Proposition 2 states that the endogenous degree of collusion is the one that maximises inefficient firm’s profits whenever firms are patient enough to sustain it as a SPNE of the repeated game. In this case, the efficient firm would rather sustain perfect collusion. However, for the inefficient firm among all the possible sustainable degrees of collusion, is the one where its profits are maximised. Therefore, is the endogenous degree of collusion. Otherwise, that is if is not sustainable, the endogenous degree of collusion is the largest sustainable one by the inefficient firm for a given δ. In other words, since collusion can only be sustained if both firms agree, the inefficient firm is the one that imposes her will. For each possible level of the discount factor, the inefficient firm is always willing to sustain a lower degree of collusion than the efficient firm, either
if firms are patient enough or
otherwise. Therefore, even though imperfect collusion will always arise in equilibrium, the degree of collusion is limited by the degree of cost asymmetry between both firms. It is also natural to analyse thus how α^{*} varies with δ and *c*.

**
Proposition 3: The endogenous degree of collusion decreases with c. Also, if
**

*, the endogenous degree of collusion increases with*δ

*.*

Intuitively, as the cost asymmetry increases, the inefficient firm is less willing to cooperate since collusion would imply a larger switch of production to the most efficient firm. Turning back to the numerical example provided above, we can better illustrate the results of the present section. Assume for instance *c* = 0.202. Then, the degree of collusion that maximises profits of the inefficient firm is
(note that if *c* = 0, obviously = 1). Therefore, if for example δ = 3, since
= 0.45 > 0.3, cannot be sustained. As a consequence, one can obtain (Proposition 2) the endogenous degree of collusion by solving the equation
= 0.3. The solution is α = 0.32 which, in this case, is the endogenous symmetric degree of collusion.

Extensions

Although some fundamental issues have been raised in the present paper, some potentially important questions still need to be addressed. In this Section, we present an alternative approach to managing the degree of cooperation, the presence of capacity constraints, and finally, we discuss the case in which firms make their strategic choices sequentially.

Considering other business practices that may possibly have anticompetitive effects and enable firms to coordinate price increases provide also a rich area for future research. For instance, ^{Holt and Scheffman (1987}) provide some interesting examples like the use of best-price policies or the public advance notification of list-price increases. In the same line, ^{García-Díaz, González and Kujal (2009}) show that, in the standard Bertrand-Edgeworth duopoly model, the use of list pricing might be a possible collusion facilitating device.

Discount factor and the degree of cooperation

We test here whether Proposition 1 hinges on the assumption made in Section 2 regarding the way to measure the intensity of competition. In particular, we assume here that the degree of cooperation is captured by the discount factor. We consider an industry like the one described in our benchmark model but for the case where α = 1. We assume also that firms play an infinitely repeated game at dates *t* = 1,...,∞ with a common discount factor
and restricting attention to the well-known grim *trigger strategies.* Each firm producing a collusive output corresponds to a SPNE if and only if the following condition is satisfied for each firm:

where ∏id denotes the one period profit from deviation and ∏ic the profits obtained by each firm at the perfect collusive equilibrium. There are many SPNE collusive output vectors that satisfy the system of inequalities in condition (6) above. As in ^{Verboven (1997}) and ^{Escrihuela-Villar (2008}), we select an equilibrium from this large set assuming that if δ exceeds a certain critical level, the set of SPNE vectors is not a binding constraint, and the distribution of output is the symmetric distribution of the output under perfect collusion. We also note that this critical level is the
previously defined evaluated at α = 1. Then, a perfectly collusive outcome is a SPNE of the repeated game if
where, as shown in the proof of Proposition 1,
is satisfied. On the contrary, if δ is below that critical level, then the set of SPNE vectors is a binding constraint, and the distribution of output is the solution to the equality constraint in (6). Unfortunately, the underlying system of equations cannot be further simplified. However, ^{Rothschild (1999}) proved that with quadratic cost functions the incentive to deviate is increasing in the deviant’s inefficiency and that, therefore, collusion is feasible whenever the most inefficient firm adheres to the agreement. Consequently, we make the simplifying assumption that we can just focus on firm 1. Let us denote by q2 the collusive output produced by firm 2. Then, it can be easily checked that the quantity produced by firm 1 in the collusive equilibrium also depends on δ. We denote it by q1 (q2, c δ) and it is given by

Notice that when δ = 0 the Cournot outcome holds, whereas
(1) the perfectly collusive equilibrium is reached. Hence, as δ varies from zero to (1) the degree of collusion increases. As the collusive profits of firm 1, that we denote by ∏_{1}
^{c}(*q*
_{2}, *δ*, *c*), depend on δ it can be proved that for all
the following is true.

**
Proposition 4: If c is small enough
** ∏

_{1}

^{c}(

*q*

_{2},

*δ*,

*c*) > ∏

_{1}

^{*},

*and*∏

_{1}

^{c}(

*q*

_{2},

*δ*,

*c*)

*increases with*δ

*. Conversely, if c is large enough*∏

_{1}

^{c}(

*q*

_{2},

*δ*,

*c*) > ∏

_{1}

^{*}

*is only true if*δ

*is low enough.*

The intuition is as follows. In a collusive equilibrium, firms are willing to cut production compared to the non-collusive equilibrium. This output reduction favours the inefficient firm as long as this firm is not “too inefficient”. Therefore, the firm is not punished to further cut its production in order to maximise joint profits. On the contrary, if firm 1 is markedly inefficient compared to firm 2, further joint profit maximisation might imply a detrimental output contraction of firm 1. Hence, Proposition 4 shows that Proposition 1 also carries over to the case where δ captures the degree of collusion.

Capacity constraints

Capacity constraints also play a key role in the analysis of tacit collusion. In our model demand is constant, so capacity constraints unambiguously affect collusion and its sustainability^{19}. It is well known that the level of firm’s capital stock determines the maximum level of production capacity, i.e. the capacity constraint. Capacity constraints affect collusion sustainability in at least two ways: they reduce the incentives to deviate as well as the severity of retaliation. Many studies on this issue have focused on symmetric situations where all firms have the same capacity (see for instance ^{Abreu, 1986}). It seems plausible to think, however, that in a model with cost asymmetries, firms’ production capacities are also asymmetric. Let us assume that firms may bear a common maximum level of cost , which is already determined by the access to capital market. Notice that there is no reason to assume that firms have different conditions to access the capital market. However, differences in efficiency may come from labour organisation and other internal production issues. In our model, this is captured by the parameter c, which in turn determines . Hence, we assume that a given level of , thus , it determines a maximum capacity level . Therefore, for any level of production q, it is hold
. As q increases both firms approach at a different path because marginal costs are higher for the inefficient firm. As a result, for a given level it can be determined a maximum level of capacity ,

where
. Thus, our model can be also interpreted as one where maximum capacity is determined by the level of c. Some studies suggest that the introduction of asymmetric capacities makes indeed collusion more difficult to sustain when the aggregate capacity is limited (see for instance ^{Davidson and Deneckere, 1990} or ^{Compte, Jenny and Rey, 2002}). The intuition is that punishing a firm with a low capacity puts an upper bound on the punishment that the other firms may inflict. Hence the other firms have to suffer the punishment they impose on the low capacity firm. Consequently, a firm with a large capacity might be reluctant to participate in such a punishment. In our model, this effect is relaxed by α. Indeed, the larger the difference in capacity, the lower α is needed to imperfectly collude. Thus, in the case of retaliation, the punishment is less severe. Hence, it seems that the introduction of asymmetric capacity constraints in our model would work in the same direction as cost asymmetries hurting also collusion sustainability. Presumably, then, a reduction in the degree of collusion could alleviate the effect of capacity constraints on collusion sustainability.

Simultaneous vs. sequential strategic choice

As ^{Mouraviev and Rey (2011}) show in a fairly general framework, such leadership is not effective in case of quantity competition since, following an aggressive deviation by the leader, the follower would rather limit its own output, making it more difficult to punish the deviation^{20}. They also show that quantity leadership along the equilibrium path does not allow the firms to achieve a Pareto improvement. The intuition is that since quantities are strategic substitutes and firms should lower their outputs to increase their profits, if the leader reduces its output then the follower should increase its own quantity in order to sustain collusion. In fact, this can also be empirically observed. In their interesting survey of EC cartel decisions, Mouraviev and Rey show that while (production or distribution) capacity appears as the key strategic variable in some cases, leadership does not feature in any of these observations.

Concluding Comments

We have developed a theoretical framework to study how firms’ cost asymmetry affects the possibility that a collusive agreement can be sustained over time. Contrary to the usual assumption made in many oligopoly models, we introduced that an imperfectly (tacit) collusive agreement can be sustained in the event that firms also care about the other firms’ profits but just to some extent. Our main contribution is twofold. We show that even though cost asymmetry hinders collusion, imperfect collusion can always be sustained regardless of the cost asymmetry. Secondly, we also analyse the endogenous degree of collusion by assuming that firms agree on a common degree of cooperation. We show that there is a limit to the degree of sustainable collusion that depends on the most inefficient firm of the industry. Another interpretation of our results is also that cost asymmetry is not necessarily a restraint for collusion as long as firms are able to sustain the maximum degree of collusion contingent on their discount factor. In this sense, some evidence suggesting collusive agreements among firms of significantly different costs of production represents an empirical justification for our findings.

The framework we have worked with is only a particular approach to a more general issue. To analyse real-world cartels, additional research is required, and for instance, a wider range of demand functions should also be considered. It would also be interesting to test if our results are robust to using an optimal punishment like the “stick-and-carrot strategies” proposed by ^{Abreu (1986}, ^{1988}). We believe that those are subjects for future research.